Resistivity Tensor of Vortex-Lattice States in Josephson Junction Arrays

Two-dimensional Josephson junction arrays frustrated by a perpendicular magnetic field are predicted to form a cascade of distinct vortex lattice states. Here, we show that the resistivity tensor provides both structural and dynamical information on the vortex-lattice states and intervening phase transitions, which allows for experimental identification of these symmetry-breaking ground states. We illustrate our general approach by a microscopic theory of the resistivity tensor for a range of magnetic fields exhibiting a rich set of vortex lattices as well as transitions to liquid-crystalline vortex states.

Figure 1

Vortex-lattice ground states for frustrations (a) $f=1/2$, (b) $f=1/3$, and (c) $f=1/5$.

Figure 2

(a)–(e) Vortex patterns for frustrations $1/3<f<1/2$. The patterns are built from a set of elementary units: a = $1/3$ (filled diagonal enclosed by empty diagonals on both sides), b = $2/5$ (two filled diagonals enclosing an empty diagonal and enclosed by an empty diagonal on both sides), c $=5/14$ (channel consisting of two $1/3$ structures enclosing a partially filled diagonal), d $=9/22$ (thick channel consisting of two $2/5$ structures enclosing a partially filled diagonal). Additional vortices are accommodated into the partially filled diagonals, changing the frustration $f$ by $1/N^2$. (f) Energy $\mathcal{E}$ per lattice site of various vortex lattice structures vs $f$, computed by gradually filling the channels with vortices. There are five color-coded regions I–V, separated by dashed lines (with the corresponding $f$ indicated). Results are limited to short periods between channels. This masks the transition at $f=2/5$, which occurs in the limit of large periods.

Figure 3

Hopping process of heavy domain wall in a $c$ structure (blue region in Fig. 2). The initial vortex configuration (dark full circles) contains a heavy domain wall (see partially filled diagonal highlighted in orange). In the adjacent saddle-point configuration, a vortex (empty circle) of the heavy domain wall hopped out of the partially filled diagonal. In the new adjacent minimum, the vortex (gray circle) returned to the partially filled diagonal, moving the heavy domain wall by two sites. We do not find minimum-energy configurations with vortices in empty diagonals.